Abstract
What is it for an imprecise credence to be justified? It might be thought that this is not a particularly urgent question for friends of imprecise credences to answer. For one might think that its answer just depends on how a well-trodden issue in epistemology plays out—namely, that of which theory of doxastic justification, be it reliabilism, evidentialism, or some other theory, is correct. I’ll argue, however, that it’s difficult for reliabilists to accommodate (the existence of justified) imprecise credences, at least if we understand such credences to be determinate first-order attitudes. If I’m right, reliabilists will have to reject imprecise credences, and friends of imprecise credences will have to reject reliabilism. Near the end of the paper, I’ll also consider whether reliabilism can accommodate indeterminate credences.
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Notes
A word on terminology: Levi (1985) reserves the term ‘imprecision’ for cases in which we have a precise credence but are unable to determine its exact value. This is not the notion of imprecision I have in mind when I talk about imprecise credences. I use the term ‘imprecise’ to talk about the lack of sharpness in an agent’s credal state itself. To talk about such lack of sharpness, Levi, as well as Hájek and Smithson (2012), use the term ‘indeterminate’. But I’ll use the term ‘indeterminate credence’ to refer specifically to what Levinstein means by ‘imprecise credence’.
For example, on this view, when we say that one is required to have a credence of [0.8, 0.9] in p, the interval [0.8, 0.9] is not meant to represent uncertainty about or indeterminacy in one’s credal attitude. Instead, it’s supposed to determinately single out a particular interval-valued credal attitude.
For more in-depth discussion of reliabilist theories of justified (precise) credence, see Dunn (2015), Tang (2016), and Pettigrew (2018). Also, since my focus is on whether reliabilism is compatible with imprecision, and not on giving a full-fledged defence of it, I’ll set aside the usual problems that reliabilism is thought to face, such as the generality problem and the New Evil Demon Problem.
Hájek (ms) argues that whereas belief is vindicated by truth, credence is vindicated by objective chance. Carr (2016) also offers a few reasons for such a view.
I discuss this theory in Tang (2016) and argue that it faces problems qua theory of justified precise credence. But here, I’ll ignore such problems—my focus is on whether there’s a natural way to extend the theory to deal with imprecision.
The Brier score is due to Glenn W. Brier (1950), who first proposed that we use it to gauge the accuracy of weather forecasts.
See, for example, Goldman (1999) (89).
Incidentally, Alston (2005) argues that his version of indicator reliabilism is also a kind of process reliabilism—he thinks that ‘reliability of process and reliability of indicator [...] coincide’ given certain plausible assumptions (137).
There’s an issue regarding what counts as a similar belief or ground. This issue is related to the generality problem—see Conee and Feldman (1998)–but we don’t need to pursue it here. Again, my aim isn’t to give a full-fledged defence of reliabilism.
Process reliabilists make a somewhat similar move when dealing with inferential justification. Reasoning by modus ponens, we infer q from our belief that p and our belief that q if p. The process of reasoning by modus ponens isn’t reliable per se since whether it yields a high ratio of true beliefs depends on whether the input beliefs are true. But the process is conditionally reliable—it produces a high ratio of true beliefs given that all the belief inputs are true. Process reliabilists might thus hold that our belief that q is justified if it is produced by a conditionally reliable process, and our beliefs that p and that q if p are themselves justified.
Fenton-Glynn (2019) argues that we should believe that imprecise objective chances exist (conditional on our adopting the Best System Analysis).
A similar problem arises if we take the Brier* score of an imprecise credence of [x, y] in p to be given by \((1 - x)^2\) if p is true and \(y^2\) if p is false. Whereas the scoring method proposed by Seidenfeld et al. (2012) systematically favours imprecision, this scoring method systematically favours precision. No matter what score we get by assigning an imprecise credence of [x, y] to p, we are guaranteed to get a lower score if we assign a precise credence of z to p, where \(x< z < y\).
For example, as Mayo-Wilson and Wheeler (2016) point out, a credence of 0.5 and a credence of [0, 1] will be equally accurate if we score an imprecise credence such as [0, 1] by its midpoint (67).
For related arguments, see Lindley (1982), Seidenfeld et al. (2012), Mayo-Wilson and Wheeler (2016), Berger and Das (2019), and Levinstein (2019). Schoenfield also argues that accuracy-first epistemology is incompatible with the position that imprecise credences are sometimes permitted. Konek (2019) tries to make room for imprecision by relating it to how much an agent values closeness to the truth and disvalues distance from error. In particular, he rejects an assumption (that of admissibility, according to which probabilistic belief states are non-dominated) that is crucial to Schoenfield’s argument for the incompatibility of accuracy-first epistemology with imprecision. On Konek’s view, ‘[w]e see different lower and upper probabilities as appropriate responses to the same evidence not because we disagree about the strength of the evidence [...] but rather because we take different attitudes toward the comparative importance of avoiding error and pinning down the truth, and different types of lower/upper probabilities (intervals) do a better job at one or the other’ (Konek 2019, 259). But as Carr (2016) notes, Konek’s ‘view does not take into account the character of an agent’s evidence and the extent to which it is informative about objective chances’—thus ignoring a major motivation for imprecision (71). For example, with respect to Black Box, our relevant evidence is that exactly 80% to 90% of the balls in the box are red. On Sturgeon’s view, any credence other than an imprecise credence of [0.8, 0.9] in our picking a red ball at random is unjustified. But on Konek’s view, different imprecise attitudes may be appropriate depending on how much we value avoiding error and value pinning down the truth. Now, ultimately, Konek might be right and Sturgeon might be wrong. But Schoenfield is explicit that her target is what she calls the Standard Imprecise View, according to which ‘if the only evidence you have concerning whether P is that the objective chance function for {P, \(\sim\)P} is in the set of probability functions S, then your evidence requires you to adopt the doxastic attitude represented by S’ (668). Further, as mentioned at the beginning of this paper, I’m assuming that Sturgeon’s intuition about Black Box is correct. This paper explores whether reliabilism can be reconciled with imprecision on such an assumption.
One might suggest that a proponent of (Grounds) should appeal to epistemic probability instead of objective probability. Suppose that there are epistemic probabilities and that such probabilities may be imprecise. Then it’s natural to suggest that your credences ought to be imprecise when the relevant epistemic probabilities are imprecise. (Comesaña (2018) endorses a theory similar to (Grounds) but maintains that the relevant notion of probability should be epistemic probability. He does not have imprecise credences in mind; instead he thinks that such a theory will avoid certain other problems faced by versions of (Grounds) that appeal to objective probability.)
However, the suggestion incurs a price that might be too hefty for most reliabilists. Reliabilists, following Goldman (1979), typically want to cash out justification in non-epistemic terms. But epistemic probability is an epistemic notion; further, it’s notoriously difficult to cash it out in non-epistemic terms. As Williamson (2000) puts it, ‘Carnap’s programme of inductive logic is moribund’, and any attempt to spell out epistemic probability in purely syntactic terms is doomed, since the ‘difference between green and grue is not a formal one’ (Williamson 2000, 211).
Also, see Rinard (2015), who provides a decision theory for imprecise credences understood as indeterminate credences.
For brevity’s sake, talk of one having an indeterminate credence of [x, y] in p should be understood as talk of it being indeterminate what precise credence one has, where such indeterminacy is represented by the interval [x, y].
Not all reliabilist theories of credences can accommodate indeterminacy easily. Recall that, according to (Probability), a justified credence in p comes from a process that produces a high proportion of credences that match the corresponding objective probabilities. Now suppose that it’s indeterminate what (precise) credence we have in RED. One might suggest that, ideally, by the lights of (Probability), it should be indeterminate what the objective probability of RED is. But this doesn’t seem right. After all, the proportion of red balls in the box before us is a determinate value—it’s just that we don’t know what it is. So, though the exact objective probability of RED is unknown, it has a determinate value too. In general, a proponent of indeterminate credences might maintain there are many cases like Black Box in which the objective probability of the relevant proposition is determinate, but it’s false that one ought to have a determinate (precise) credence in that proposition. (I’ll leave it to the reader to think about whether (Calibration) and (Brier) can accommodate indeterminate credences.)
The expression ‘forming an indeterminate credence of [0.8, 0.9] in p’ should be interpreted in the way mentioned in footnote 21.
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Acknowledgements
I thank Ben Blumson, Michael Pelczar, and Jens Christian Bjerring for comments on an early version of this paper. I also thank the participants of an epistemology workshop at the National University of Singapore, the audience at the 2014 International Conference on Epistemology and Cognitive Science at Xiamen University, the participants of a reading group organised by Richard Pettigrew at the University of Bristol, and the audience at a King’s College London seminar organised by Julien Dutant. Last but not least, I thank an anonymous referee of this journal for their helpful feedback.
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Tang, W.H. Reliabilism and imprecise credences. Philos Stud 178, 1463–1480 (2021). https://doi.org/10.1007/s11098-020-01491-2
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DOI: https://doi.org/10.1007/s11098-020-01491-2